 3.2.1: Determine whether the following sets form subspaces of R2: (a) {(x1...
 3.2.2: Determine whether the following sets form subspaces of R3: (a) {(x1...
 3.2.3: Determine whether the following are subspaces of R22: (a) The set o...
 3.2.4: Determine the null space of each of the following matrices: (a) 2 1...
 3.2.5: Determine whether the following are subspaces of P4 (be careful!): ...
 3.2.6: Determine whether the following are subspaces of C[1, 1]: (a) The s...
 3.2.7: Show that Cn[a, b] is a subspace of C[a, b].
 3.2.8: Let A be a fixed vector in Rnn and let S be the set of all matrices...
 3.2.9: In each of the following, determine the subspace of R22 consisting ...
 3.2.10: Let A be a particular vector in R22. Determine whether the followin...
 3.2.11: Determine whether the following are spanning sets for R2: (a) _ 2 1...
 3.2.12: Which of the sets that follow are spanning sets for R3? Justify you...
 3.2.13: Given x1 = 1 2 3 , x2 = 3 4 2 , x = 2 6 6 , y = 9 2 5 (a) Is x Span...
 3.2.14: Let {x1, x2, . . . , xk } be a spanning set for a vector space V. (...
 3.2.15: In R22, let E11 = 1 0 0 0 , E12 = 0 1 0 0 E21 = 0 0 1 0 , E22 = 0 0...
 3.2.16: Which of the sets that follow are spanning sets for P3? Justify you...
 3.2.17: Let S be the vector space of infinite sequences defined in Exercise...
 3.2.18: Prove that if S is a subspace of R1, then either S = {0} or S = R1. 1
 3.2.19: Let A be an n n matrix. Prove that the following statements are equ...
 3.2.20: Let U and V be subspaces of a vector space W. Prove that their inte...
 3.2.21: Let S be the subspace of R2 spanned by e1 and let T be the subspace...
 3.2.22: Let U and V be subspaces of a vector space W. Define U + V = {z  z...
 3.2.23: Let S, T, and U be subspaces of a vector space V. We can form new s...
Solutions for Chapter 3.2: Subspaces
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 3.2: Subspaces
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 23 problems in chapter 3.2: Subspaces have been answered, more than 4275 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 3.2: Subspaces includes 23 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).